integral of (2x-3)/(x^3+x)

Lösung

\int \frac{2 x - 3}{x ^{3} + x} d x

2 arctan (x) - 3 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1) + C

Schritte zur Lösung

\int \frac{2 x - 3}{x ^{3} + x} d x

Multipliziere aus

\frac{2 x - 3}{x ^{3} + x} : \frac{2 x}{x ^{3} + x} - \frac{3}{x ^{3} + x}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{2 x}{x ^{3} + x} d x - \int \frac{3}{x ^{3} + x} d x

\int \frac{2 x}{x ^{3} + x} d x = 2 arctan (x)

\int \frac{3}{x ^{3} + x} d x = 3 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1)

= 2 arctan (x) - 3 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1)

Füge eine Konstante zur Lösung hinzu

= 2 arctan (x) - 3 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1) + C